Show that a ring R is commutative if and only it a - b = (a+ b) (a - b) for all a, be R.

Q: Let R = {2n: n E Z} and define addition O and multiplication O in R by a.b a O b : = a + b and aOb…

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Q: (a) Let R be a ring and S a subset of R. What does it mean to say that S is a subring of R?

A: a. S is a subset of R. A non-empty subset S of R is a subring if a, b ∈ S ⇒ a - b, ab ∈ S. A subring…

Q: The ring 5Z is isomorphic to the ring 6Z OTrue O False

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Q: Give an example of a non-commutative ring R without unity such that (xy)^2 = x^2.y^2 for all x,y in…

A: We consider the example of a non-commutative ring R without unity such that (xy)2 = x2y2 for all x…

Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ....... O…

A: Here you have posted multiple question, So as per the policy I can answer only first question for…

Q: If R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…

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Q: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

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Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

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Q: Show that a ring is commutative if it has the property that ab = caimplies b = c when a ≠ 0.

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Q: Let a and b be elements of a ring. Prove that (-a)b = -(ab).

A: Solve the following

Q: Let 1={0,2} CZ. Prove if ring Z,/l is isomorphic with a ring Z,

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Q: a is a unit in a ring R with unity, then a is not a zero divisor

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Q: If in a ring R every x E R satisfies x2 = x, Prove that R must be commutative.

A: Answer and explanation is given below...

Q: Let R = {2n: n E Z} and define addition and multiplication O in R by a b = a + b and aOb = for all…

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Q: The ring Z is isomorphic to the ring 3Z O True False

A: Solution:

Q: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all…

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Q: If R is a ring, then every element of R is either a unit or a zero-divisor. Select one: O True O…

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Q: Let U = {a, b}. Define addition and multiplication in P(U) by C +D = CU D and CD = Cn D. Decide…

A: Ring (definition) Let R be a non empty set together with two binary operations called addition(+)…

Q: Find elements a, b, and c in the ring Z ⨁ Z ⨁ Z such that ab, ac, andbc are zero-divisors but abc is…

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Q: The ring Z is isomorphic to the ring 3Z False True

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Q: The measure u is monotone on the ring. So that µ(A) < µ(B) if ACB

A: Given that The measure is monotone ob the ring,  So we need to consider the following;

Q: The ring 5Z is isomorphic to the ring 6Z False True

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Q: 5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element…

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Q: The ring 5Z is isomorphic to the ring 6Z

A: Since the z is same for both and the multiplier is different i.e. 5 &amp;6.

Q: Let CR,t,,) be a Commutative ring ?

A: First I recall you Definition of commutative ring.  A commutative ring is ring in which…

Q: The set of all units of the ring Zg is

A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0

Q: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

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Q: 24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different…

A: R, +,· is said to be commutative ring if Suppose R is a non empty set such that for any two elements…

Q: Let R, , O is a ring under two composion e and O üs follows ü e i; = a + b + 1 and aOb = ab + a + b…

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Q: Let R be a ring and a=a for all a'e R, Then commutative. prove that R is

A: First we notice that x3=x for all x∈ℝ, so that means 2x3=2x and thus 8x=8x3=2x and so 6x=0. Thus…

Q: The ring 5Z is isomorphic to the ring 6Z True O False

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Q: The set of all idempotents of the ring Z is اختر احدى الاجابات O (1,0) O (0,1) O (0,-1,1} O (1,-1)

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Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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Q: The ring 3z is isomophic to the ring 5Z False True

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Q: Let R be a ring and assume a∈R is not a zero divisor.Prove that if ba=ca,  then b=c.

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Q: Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a…

A: We have to solve given problem:

Q: If Ris a ring with identity and a is a unit, prove that the equation ax = has a unique solution in…

A: Let R be a ring with identity and a∈R be a unit. Prove that the equation ax=b has a unique solution…

Q: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

Q: Find two elements a and b in a ring such that both a and b are zerodivisors,a + b ≠ 0, and a + b is…

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Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

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Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…

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Q: Prove that Q[x]/&lt;x2 - 2&gt; is ring-isomorphic to Q[√2] = {a +b√2 | a, b ∈ Q}.

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Q: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).

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Q: Let R be a commutative ring with unity and r ∈ R. Prove that if ⟨r⟩ = R, then r is a unit. Consider…

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Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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Q: Let R be a ring and M be an R-module. Ther

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Q: An element x in a ring is called an idempotent if x2 = x. Prove that  the characterstic of R is 0 or…

A: An element x in a ring is called an idempotent if x^2 = x

Q: The ring 3z is isomorphic to the ring 5z True False

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Q: The number of zero divisors of the ring Z, O Zg is

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